The corresponding user page gives only a 404, as well as any moderator-only pages about this user.

After some brainstorming with other moderators, we were able to reproduce this: An unregistered user has a delete link in his profile, which seems to delete the cookies (or the server-side association with this cookies). This link does even work when this user has an upvoted question or answer, as we (me and Gilles) tested as user1120 and user1121 on this other now deleted question.

1 Answer
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Users only get to self delete if they have at most ONE post or vote, deleted or not (we may adjust the deleted or not check in the future, not planned right now though). This check happens when showing the link and when handling the deletion endpoint, you cannot bypass it.
Since a bounty is a type of vote under the covers, it falls under this check as well.

However this did alert us to the fact we forgot to add the self delete link to the new user profile...this will happen in the next build.